On the error incurred using the bootstrap variance estimate when constructing confidence intervals for quantiles

We show that the coverage error of confidence intervals and level error of hypothesis tests for population quantiles constructed using the bootstrap estimate of sample quantile variance is of precise order n-1/2 in both one- and two-sided cases. This contrasts markedly with more classical problems, where the error is of order n-1/2 in the one-sided case, but n-1 in the two-sided case, and results from an unusual feature of the Edgeworth expansion in that the leading term, of order n-1/2, is proportional to a polynomial containing both odd and even powers of the argument. Our results also show that for two-sided confidence intervals and hypothesis tests, and in large samples, the bootstrap variance estimate is inferior to the Siddiqui-Bloch-Gastwirth variance estimate provided the smoothing parameter in the latter is chosen to minimize coverage/level error.